Last updated on July 4th, 2025
Cumulative frequency is the progressive total of frequencies in a data set. The total data is arranged in a table where the frequency is divided according to their class intervals. In this article, we will learn more about cumulative frequency and its types.
When we get the running sum of frequencies in a given dataset, we call it cumulative frequency. The frequency of the first class interval is added to the second frequency of the second class interval and so on, until the last frequency.
The cumulative frequency shows how many values in a dataset fall at or are below a particular value. It accumulates the frequency of data as you move through the dataset.
To calculate cumulative frequency, we take the frequency of the first class interval, then we add it to the frequency of the second class interval, and so on. The cumulative frequency is calculated using the formula:
Where:
Below are the steps to calculate the cumulative frequency:
Step 1: First we sort the data and arrange it in a table
Step 2: Then we calculate the frequency of each value in the dataset
Step 3: The next step is to calculate the cumulative frequency
Step 4: The cumulative frequency of the first class interval is the same as the frequency of the first class interval
Step 5: Find the cumulative frequency of the next class interval
Step 6: Repeat for the remaining class intervals
Step 7: Double-check your answers to avoid careless errors
Below is a table explaining how cumulative frequency is calculated.
Month | Frequency (number of toys sold) | Cumulative frequency (total number of toys sold) |
January | 50 | 50 |
February | 60 | 50 + 60 = 110 |
March | 70 | 110 + 70 = 180 |
April | 80 | 180 + 80 = 260 |
You can see that the last cumulative frequency is equal to the total of all observations. This is true for the final cumulative frequency.
Cumulative frequencies are categorized into two types:
Less than cumulative frequency, also known as less than ogive. It is obtained by adding the frequency of the first class interval to the frequency of the second-class interval and so on. Here, the cumulative frequency begins from the lowest class to the highest class. In a graph, less than cumulative frequency is shown as a rising curve.
More than cumulative frequency is calculating by the cumulative frequency from the last class to the first class. In this method, we start the cumulative frequency from the highest to the lowest class. In a graph, more than cumulative frequency is drawn as a downward curve.
To plot the points in a graph we use the cumulative frequency. To draw a cumulative graph (also called ogive, follow these steps:
Step 1: Create a cumulative frequency table
Score | Frequeny | Cumulative Frequency |
0 - 10 | 2 | 2 |
10 - 20 | 5 | 2 + 5 = 7 |
20 - 30 | 8 | 7 + 8 = 15 |
30 - 40 | 6 | 15 + 6 = 21 |
40 - 50 | 4 | 21 + 4 = 25 |
Step 2: Identify the scales of the graph
Here, in the x-axis we represent the scores and the y-axis represents the cumulative frequency.
The x-axis would be 10, 20, 30, 40, 50 and the y-axis would be 0, 5, 10, 15, 20, 25.
Step 3: Plot the points in the graph.
Here the points are:
Step 4: Connect the points in the graph to complete the ogive.
We can use three methods to graphically represent cumulative frequency data:
Cumulative Frequency Curve - In this method, we will be creating the graph by plotting cumulative frequencies against the upper class boundaries of the dataset. We then use a smooth curve to connect the points.
Here is a cumulative frequency curve for better understanding:
Cumulative Frequency Polygon: A line graph connecting cumulative frequencies at class midpoints.
Here is the cumulative frequency polygon using the example of score
Cumulative Frequency Graph: It can be represented as any kind of graph, even a bar graph showing the cumulative frequency.
Here is a cumulative frequency graph using the above example
When calculating cumulative frequency and plotting graphs students may get confused and make mistakes. So here are a few mistakes that students make and ways to avoid them.
Cumulative frequency is widely used in the real world. It helps us understand how data is accumulated over a period of time. Here are a few real-world applications of cumulative frequency.